Lemma 6.3.18. Let \(C\) be a stable \(\infty \)-category equipped with a t-structure, and let \(X \in C_{\geq 0}\) be a connective object. Then \(X\) lies in the heart of \(C\) if and only if it is a 0-truncated object of \(C_{\geq 0}\), in the sense that \(\pi _k\Hom _{C_{\geq 0}}(Y,X) = 0\) for all \(Y \in C_{\geq 0}\) and \(k > 0\).
Proof. First assume that \(X \in C^{\heartsuit }\). Then for \(Y \in C_{\geq 0}\) and \(k > 0\) we have \(\pi _k\Hom _C(Y,X) \simeq \pi _0\Hom _C(Y,X[-k]) = 0\) as \(X[-k] \in C_{\leq -k} \subseteq C_{\leq -1}\). Conversely, if \(X\) is 0-truncated in \(C_{\geq 0}\), then we have \(\pi _k\Hom _C(Y,X[-1]) \cong \pi _{k+1}\Hom _C(Y,X) \cong 0\) for all \(Y \in C_{\geq 0}\) and \(k \geq 0\). It follows that \(X[-1] \in C_{\leq -1}\), and thus \(X \in C^{\heartsuit }\). □
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